THE CONDITION FOR AN ORTHONOMIC
DIFFERENTIAL SYSTEM*
BY
JOSEPH MILLER THOMAS
This paper develops a method of testing whether a given finite set of
partial derivatives can be placed in a specified order by assigning integral
cotes to the independent variables and functions in Riquier's manner, f The
method gives a test for determining whether a system of partial differential
equations is orthonomic, as far as the ordering of derivatives is concerned.
1. Consider r functions ua of n independent variables x¡. The partial de-
rivative
(1.1)
Qil+ •••+<»«„
£,«« = —-r
dx\l ■ ■ ■ dxn"
is conveniently represented by a matrix whose elements are integers
which has one row and n+r columns, namely, by
(1.2)
and
/ = ||il»l ■ • • 1.0 ■ • ■•1 • • -0||,
the element 1 being in the («+a)th
column.
Riquier's method of placing the partial derivatives of the functions « in a
definite order can be described as follows. Let there be given a fixed matrix M
whose elements are integers and whose rows are n+r in number. The elements on the ath column are called the ath cotes of the variables u, x. Let J
be the matrix associated with another derivative, say with
(1.3)
v
d'i+---+'"uß
DjUß =- dx" ■■ ■dx„« •
By definition the derivative I follows or precedes J according as the first nonzero element in the matrix (of one row)
(1.4)
(I-J)M
is positive or negative. This statement may be abbreviated
saying that / follows or precedes / according as
symbolically
(I - J)M > 0 or (I - J)M < 0.
* Presented to the Society, October 31, 1931; received by the editors in September, 1931.
t Riquier, C, Les Systèmes d'Equations aux Dérivées Partielles, Paris, 1910, p. 195.
332
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by
333
ORTHONOMIC DIFFERENTIAL SYSTEMS
Suppose there are given certain order relations among a finite set of derivatives of the m's. Each of these order relations can be given the form "7
follows J." The problem of determining a matrix M which establishes a given
set of order relations among the derivatives can accordingly be phrased as
follows: find a matrix of integers M which satisfies the inequalities
(1.5)
KM > 0,
where K( —I—J) assumes a finite set of given values.
The conditions on the elements Xi of the first column of M are
n+r
(1.6)
£Xi"^èO,
p-1
where
kp = ip —jp
kn+a = 1, kn+ß = — 1
(p = 1, 2, ■■• , n)^
(a 9¿ ß),
and all the other k's are zero. If a =ß, then kn+x= • • • = kn+T= 0.
The discussion of system (1.6) is most readily accomplished, it seems, by
adopting the geometric point of view employed by Miss Stokes.* To each of
the given relations there corresponds a point in (n+r) -dimensional euclidean
space, whose coordinates are (kx, k2, ■ ■ ■, k„+r). We shall speak of the set of
points corresponding to the given order relations as the "set of representative
points Sn+r" A solution of (1.6) is an oriented (n+r— l)-flat passing through
the origin and separating no pair of representative points k.
There is a sub-set of Sn+r, called its inconsistent set, contained by every
solution of (1.6).t An important property of the inconsistent set is given by
Theorem
1. The system of inequalities
n+r
(1.7)
Ea'î,
p-i
â 0
whose coefficients are the coordinates of the points in any inconsistent set has only
equality solutions, that is, solutions making all its left members zero.
To prove the above, we remark that the general solution of the system in
question is given byî
* Stokes, R. W., A geometric theory of solution of linear inequalities, these Transactions, vol. 33
(1931),pp. 782-805.
f Stokes, loc. cit., p. 794. When we use the unqualified term "inconsistent set," we mean the
case 1= 0, that is, the solution is not required a priori to contain any points of the original set. From
(3.3) of that paper it is moreover clear that Theorem 9 is true whatever the rank of the system.
Î Stokes, loc. cit., p. 786, formula (3.3).
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334
J. M. THOMAS
(1.8)
0-+
[April
£a*«i,
<=«+i
where q is the dimensionality of the flat space determined by the inconsistent
'set and the origin, the «'s are expressions whose vanishing defines that flat
space, and a is the general solution of the system determined by the point set
in q dimensions. From the definition of an inconsistent set,* however, a = 0.
Hence the (n+r —l)-flat (1.8) passes through all the points of the inconsistent set because the coordinates of those points make the u's zero. Consequently the only solutions of (1.7) are equality solutions, and the theorem is
demonstrated.
In passing, it is perhaps worth while to mention the following corollary :
Theorem
2. The inconsistent set of an inconsistent set of points is the set
itself.
Returning to the discussion of the determination
(1.5), we see that the second cotes Xf must satisfy
of a matrix satisfying
n+r
(1.9)
¿Z^kp^O,
p-i
where (kx, ■ ■ • , kn+T) ranges over the inconsistent set of 5„+r. This is true
because any solution of (1.6) passes through the inconsistent set of Sn+r,
that is, the left members of (1.6) which correspond to the inconsistent-set are
zero for every solution of (1.6). The first elements in the corresponding KM's
being zero, it is necessary that the second elements be non-negative.
Theorem 1 shows that (1.9) has only equality solutions. The third cotes
must therefore satisfy the same system as the second. This statement is true
of all succeeding cotes. Consequently we have
KM = 0,
where K ranges over the inconsistent set, no matter how many columns of M
are determined to satisfy (1.6) and the analogous succeeding conditions.
If the inconsistent set actually contains one or more points, it is therefore
impossible to determine an M placing the derivatives in the specified order.
If the inconsistent set is vacuous, the associated system
n+r
(1.10)
£Xr°^>0,
p-1
obtained from (1.6) by replacing the sign ^ by >, has a solution.f
* Stokes, loc. cit., p. 794, for 1=0.
f Stokes, loc. cit., p. 794, Theorem 10 for 1= 0.
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1932]
ORTHONOMIC DIFFERENTIAL SYSTEMS
335
When the rank of (1.10) is n+r, its general solution is a linear homogeneous combination of the complete system of fundamental solutions of (1.6),
the coefficients being arbitrary positive constants.* A fundamental solution
is obtained from the given numbers k by rational operations, and is therefore
composed of rational numbers, when the k's are integral, as they are here.
The arbitrary constant, which multiplies any fundamental solution of (1.6)
as it appears in the general solution of (1.10), can be chosen so as to remove
the denominator occurring in the fundamental solution. Hence, *'/ (1.10) has
a solution, it has a solution in integers.
The case where the rank of (1.10) is less than n+r can be reduced to the
case already treated.f The result just established therefore holds in general.
We have accordingly shown that if a matrix establishing the given order
relations exists, system (1.10) has a solution in integers Xi. A matrix with a
single column composed of these X's will effect the given ordering. Hence we
have
Theorem 3. If a finite set of order relations can be established by a matrix
of integers, it can be established by a matrix of integers having a single column. I
As a consequence of this and of the geometric condition § that (1.10) have
a solution we get
Theorem 4. A given finite set of order relations can be effected by a matrix
of integers if and only if the origin is exterior to the convex figure determined by
the representative points.
Analytic methods for testing data and for finding the X's when they exist
are to be found in Miss Stokes' paper.
2. Riquierll finds it desirable to make the first cotes of the independent
variables unity, i.e., to make the first n elements on the first column of M
equal to unity. If we let
k = ix +
the remaining
■■ ■+
i„ — jx — ■ ■ ■ — jn,
first cotes pn+1, ■ • • , pn+T must satisfy
* Stokes, loc. cit., p. 793, Theorem 8.
t Stokes, loc. cit., p. 786. If the a's in (3.3) are made zero, it is clear that the general solution
in the subspace is a particular solution of the original system.
| The theorem is not true in general if the set of order relations is infinite. For example, a matrix
M ordering all derivatives must contain at least n columns, where n is the number of independent
variables. Cf. a paper by the author, Matrices of integers ordering derivatives, these Transactions,
vol. 33 (1931),p. 393.
§ Stokes, loc. cit., p. 804, Theorem 16.
|| Loc. cit., p. 207, footnote 2.
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^36
[April
J. M. THOMAS
n+r
(2.1)
k+
JZ^kp^Q.
V—n+l
This system has a solution if and only if the homogeneous
system
n+r
(2.2)
pk+
£
p»kp 5; 0, p > 0,
J>=n+1
has a solution.*
The points ik, kn+x, ■ • ■, kn+r) plus the point (1, 0, • • • , 0) will be referred to as the "setSr+i of representative points in r+1 dimensions" just
as the set previously described is the "set 5„+r of representative points in
n+r dimensions." System (2.2) has a solution if and only if
n+r
(2.3)
pk+
£ n'k, è 0, p è 0,
î>—n+l
has a solution not containing the point (1,0, • • • ,0). Since the general solution of (2.3) contains the inconsistent set of S r+i, it is necessary that
(1, 0, ■ ■ • , 0) be not in that inconsistent set. We suppose this condition
fulfilled.
The only points of S r+i which are in all the fundamental solutions of (2.3)
are those of its inconsistent set.f Hence by giving positive values to the arbitrary constants in the general solution of (2.3) we get a solution of (2.2) which
contains no point of S r+i except those in its inconsistent set.
Let the points of Sn+r which correspond to points in the inconsistent set
of S r+i be called the derived set of Sn+r and denoted by Sn+r- When the first
cotes have the values whose determination was indicated above, in (1.6) the
sign = holds for points of Sñ+r and the sign > for all other points. Hence
the only conditions to be satisfied by the second cotes are (1.9) in which k
ranges over the set 5„'+r. The discussion concerning the existence of Xi's satisfying (1.6) in the proof of Theorem 3 applies verbatim.
As before, the operations in solving are rational. The elements of M can
be rendered integral by multiplying them by a properly chosen positive integer. Hence we have the following results.
Theorem 5. 7/ a given finite set of order relations among derivatives can be
established by means of a matrix of integers, the first cote of each independent
variable being unity, it can be established by a matrix of two columns.
* Stokes, loc. cit., §13.
t This result follows readily from Theorem 11 of Miss Stokes' paper.
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1932]
ORTHONOMIC DIFFERENTIAL SYSTEMS
337
Theorem 6. A given finite set of order relations can be effected by a matrix
of integers having unity as the first cote of each independent variable if and only
if the point (1, 0, • • • , 0) is not in the inconsistent set for r+l dimensions and
the origin is exterior to the convex figure determined by the derived set of points
in n+r dimensions.
The above method can be employed to obtain a necessary and sufficient
condition for the existence of a matrix which establishes a given set of order
relations and which has any set of its elements given.
3. In applying the foregoing results, the dimensionality of the spaces considered can be diminished by unity in the following manner. The elements of
M corresponding to any function u, that is, the elements on any one of the
last r rows, can be made zero by a transformation of M which preserves order*
Hence, in particular, the elements in the last row of M can be assumed as
zero, and consequently the last coordinate of the points in 5n+r and S r+i ig-
nored.
4. The chief application of the above is in determining whether a given
system of partial differential equations is orthonomic.f To do this, it is necessary to determine whether there is a matrix of integers placing the derivatives
in such an order that all the derivatives appearing in any given right member
precede the derivative which constitutes the corresponding left member.
This application will now be illustrated by two examples. First consider
the single equation í
d2u
(4.1)
( d2u
-dxdy = F(-,
\dx2
d2u\
-V
dy2/
In order that the system be orthonomic, it is necessary that
d2u
d2U
> -,
dxdy
dx2
-
-
d2U
dxdy
>-,
d2U
dy2
the sign > being read "follows." These conditions are represented
geometri-
cally by the points
S3:(-
I, 1,0), (1, - 1,0);
S2: (0,0), (0,0), (1,0).
The inconsistent set of S 2 is thus (0, 0), and Sí coincides with S3. The convex
figure determined by Sí is a line segment containing the origin. Hence (4.1)
* Thomas, loc. cit., p. 391.
t Riquier, loc. cit., p. 201.
Î Riquier, loc. cit., p. xx.
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338
J. M. THOMAS
is not orthonomic. It will become orthonomic
from its right member.
As a second example consider the system
du
(d2v
if either derivative
is removed
\
Yx~ W2' 7'
(4.2)
The representative
d3v
(d2u
d2u\
dx2dy
\dy2
dxdy)
points are
S4 : (1, - 2, 1, - 1), (1, 0, 1, - 1), (2, -1,-1,
1), (1, 0, - 1, 1);
Sa : (- 1, 1, - 1), (1, 1, - 1), (1, - 1, 1), (1, - 1, 1), (1, 0, 0).
Ignoring the last coordinate, we find the inconsistent set of S3 to be ( —1, 1)
and (1, —1). Since (1, 0) is not in the inconsistent set, the first condition of
Theorem 6 is satisfied. Furthermore, the derived set Si is seen to be
Si : (1, - 2, 1), (2, - 1, - 1),(1,0,-1).
The origin is exterior to the convex figure determined by these three points.
Hence system (4.2) is orthonomic, as far as ordering of derivatives is concerned. A matrix putting the derivatives in the desired order is
M =
1
1
1
0
1
0
0
0
Düke University,
Durham, N. C.
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